Randomized benchmarking by exploiting the structure of the Clifford group

ABSTRACT

A method of generating a randomized benchmarking protocol includes providing a randomly generated plurality of Hadamard gates; applying the Hadamard gates to a plurality of qubits; and generating randomly a plurality of Hadamard-free Clifford circuits. Each of the plurality of Hadamard-free Clifford circuits is generated by at least randomly generating a uniformly distributed phase (P) gate, and randomly generating a uniformly distributed linear Boolean invertible matrix of conditional NOT (CNOT) gate, and combining the P and CNOT gates to form each of the plurality of Hadamard-free Clifford circuits. The method also includes combining each of the plurality of Hadamard-free Clifford circuits with corresponding each of the plurality of Hadamard gates to form a sequence of alternating Hadamard-free Clifford-Hadamard pairs circuit to form the randomized benchmarking protocol; and measuring noise in a quantum mechanical processor using the randomized benchmarking protocol.

BACKGROUND

The currently claimed embodiments of the present invention relate toquantum computation, and more specifically, to methods of generating arandomized benchmarking protocol.

Clifford circuits can be used in quantum computing for quantum errorcorrection and in fault-tolerant computations. Optimization of Cliffordcircuits can be useful within the scope of fault-tolerant computations,as the Clifford overhead can dominate the cost of an implementation,despite the cost of the non-Clifford gates being higher than that of theClifford gates. Clifford circuits can also be used in the study ofentanglement and in the study of noise in quantum computers viarandomized benchmarking (RB). Random uniformly distributed Cliffordcircuits have many applications in quantum computing including inrandomized benchmarking protocols, randomized quantum code construction,quantum data hiding, and in compressed classical description of quantumstates.

Randomized benchmarking (RB) is an efficient and robust method tocharacterize gate errors in quantum mechanical circuits. Averaging overrandom sequences of gates can provide estimates of gate errors in termsof the average circuit fidelity. These estimates are independent fromthe state preparation and measurement errors that plague other methodssuch as channel tomography and direct fidelity estimation. Standard RBprotocols consist of repeated application of a sequence of k randomn-qubit Clifford circuits, C1C2 . . . Ck, followed by Ck+1 that performsthe inversion of the unitary transformation obtained by the circuit C1 .. . Ck and measuring the result.

SUMMARY

An aspect of the present invention is to provide a method of generatinga randomized benchmarking protocol. The method includes providing arandomly generated plurality of Hadamard gates; and applying theHadamard gates to a plurality of qubits; and generating randomly aplurality of Hadamard-free Clifford circuits. Each of the plurality ofHadamard-free Clifford circuits is generated by at least randomlygenerating a uniformly distributed phase (P) gate, and randomlygenerating a uniformly distributed linear Boolean invertible matrix ofconditional NOT (CNOT) gate, and combining the P and CNOT gates to formeach of the plurality of Hadamard-free Clifford circuits. The methodfurther includes combining each of the generated plurality ofHadamard-free Clifford circuits with corresponding each of the randomlygenerated plurality of Hadamard gates to form a sequence of alternatingHadamard-free Clifford-Hadamard pairs circuit to form the randomizedbenchmarking protocol; and measuring noise in a quantum mechanicalprocessor using the randomized benchmarking protocol.

In an embodiment, generating randomly the plurality of Hadamard gatesincludes generating the plurality of Hadamard gates H_(1 . . . k), wherek is an integer greater than 2 and corresponds to a number of randomqubits sampled from n-qubits, n being greater than or equal to 2.

In an embodiment, generating randomly the plurality of Hadamard-freeClifford circuits includes generating the plurality of Hadamard-freeClifford circuits F_(1 . . . k), where k is an integer greater than 2and corresponds to a number of random qubits sampled from n-qubits, nbeing greater than or equal to 2.

In an embodiment, combining each of the generated plurality ofHadamard-free Clifford circuits with corresponding each of the randomlygenerated plurality of Hadamard gates to form the sequence ofalternating Hadamard-free Clifford-Hadamard pairs circuit includesforming the sequence F₁H₁ . . . F_(k)H_(k)F_(k+1)H_(k+1), where k is aninteger. In an embodiment, the plurality of Hadamard-free Cliffordcircuits are different from each other and are chosen at randomaccording to a specified distribution.

In an embodiment, generating each of the plurality of Hadamard-freeClifford circuits includes randomly generating the uniformly distributedphase (P) gate, randomly generating the uniformly distributed linearBoolean invertible matrix of conditional NOT (CNOT) gate, and randomlygenerating a uniformly distributed conditional rotation around axis-Z(CZ) gate, and combining the P, CZ, and CNOT gates to form each of theplurality of Hadamard-free Clifford circuits.

In an embodiment, combining each of the generated plurality ofHadamard-free Clifford circuits with corresponding each of the randomlygenerated plurality of Hadamard gates to form a sequence ofHadamard-free Clifford-Hadamard pairs circuit includes forming thesequence F₁H₁ . . . F_(k)H_(k)F_(k+1)H_(k+1), where k is an integer,wherein: the Hadamard-free Clifford circuits F₁, F₂, . . . , F_(k) areequal to a product P.CZ.CNOT, where P is a uniformly distributed phase(P) gate, CZ is a uniformly distributed conditional rotation aroundaxis-Z (CZ) gate, and CNOT is a uniformly distributed linear Booleanconditional NOT (CNOT) circuit, and the Hadamard-free Clifford circuitF_(k+1) is reduced to contain no more than nk−k(k+1)/2 two-qubit CNOTand CZ gates, where k is the number of Hadamard gates in Bruhatexpansion of F₁H₁ . . . F_(k)H_(k), where nk−k(k+1)/2 is obtained byreducing a left-hand Hadamard-free element by commuting and merginggates in it to the right.

In an embodiment, a computational complexity of generating therandomized benchmarking protocol grows quadratically with a number ofqubits. In an embodiment, measuring the noise in the quantum mechanicalcircuit includes combining an additional Hadamard-free Clifford-Hadamardpair to the formed sequence of Hadamard-free Clifford-Hadamard pairscircuit to perform an inversion of a unitary transformation obtained bythe formed sequence of Hadamard-free Clifford-Hadamard pairs circuit.

In an embodiment, the method further includes optimizing a length of therandomized benchmarking protocol by using fewer gates in the pluralityof Hadamard-free Clifford circuits than a number of circuit elements ina full Clifford circuit. In an embodiment, a size of the plurality ofHadamard-free Clifford circuits are computed using the followingequation:

${{\mathcal{F}(n)}} = {2^{{2n} + n^{2}}{\prod\limits_{j = 1}^{n}\;\left( {2^{j} - 1} \right)}}$

and the number of circuit elements in the full Clifford circuit iscomputed using the following equation:

${{\mathcal{C}(n)}} = {2^{{2n} + n^{2}}{\prod\limits_{j = 1}^{n}\;\left( {4^{j} - 1} \right)}}$

where n is a number of qubits, j is an independent variable and j is anincrement from 1 to n number of qubits.

Another aspect of the present invention is to provide a computerreadable medium on which is stored non-transitory computer-executablecode, which when executed by a classical computer causes a quantumcomputer to: (a) provide a randomly generated plurality of Hadamardgates; (b) apply the randomly generated Hadamard gates to a plurality ofqubits; (c) generate randomly a plurality of Hadamard-free Cliffordcircuits, wherein each of the plurality of Hadamard-free Cliffordcircuits is generated by at least randomly generating a uniformlydistributed phase (P) gates, and randomly generating a uniformlydistributed linear Boolean invertible matrix of conditional NOT (CNOT)gates, and combining the P and CNOT gates to form each of the pluralityof Hadamard-free Clifford circuits; (d) combine each of the generatedplurality of Hadamard-free Clifford circuits with corresponding each ofthe randomly generated plurality of Hadamard gates to form a sequence ofalternating Hadamard-free Clifford-Hadamard pairs circuit to form therandomized benchmarking protocol; and (e) measure noise in a quantummechanical processor using the randomized benchmarking protocol.

In an embodiment, the non-transitory computer-executable code whenexecuted by the classical computer causes the quantum computer togenerate the plurality of Hadamard gates H_(1 . . . k), where k is aninteger greater than 2 and corresponds to a number of random qubitssampled from n-qubits, n being greater than or equal to 2.

In an embodiment, the non-transitory computer-executable code whenexecuted by the classical computer causes the quantum computer togenerate the plurality of Hadamard-free Clifford circuits F_(1 . . . k),where k is an integer greater than 2 and corresponds to a number ofrandom qubits sampled from n-qubits, n being greater than or equal to 2.

In an embodiment, the non-transitory computer-executable code whenexecuted by the classical computer causes the quantum computer tocombine each of the generated plurality of Hadamard-free Cliffordcircuits with corresponding each of the randomly generated plurality ofHadamard gates to form the sequence F₁H₁ . . . F_(k)H_(k)F_(k+1)H_(k+1),where k is an integer. In an embodiment, the plurality of Hadamard-freeClifford circuits are different from each other and are chosen at randomaccording to a specified distribution.

In an embodiment, the non-transitory computer-executable code whenexecuted by the classical computer causes the quantum computer to applythe uniformly distributed phase (P) gates, apply the uniformlydistributed random linear Boolean invertible matrix computed byconditional NOT (CNOT) gates, and apply a random uniformly distributedconditional rotation around axis-Z (CZ) gate, and combine the P, CZ, andCNOT gates to form each of the plurality of Hadamard-free Cliffordcircuits.

In an embodiment, the non-transitory computer-executable code whenexecuted by the classical computer causes the quantum computer tooptimize a length of the randomized benchmarking protocol by using fewergates in the plurality of Hadamard-free Clifford circuits than a numberof circuit elements in a full Clifford circuit.

In an embodiment, the number of elements of the plurality ofHadamard-free Clifford circuits are computed using the followingequation:

${{\mathcal{F}(n)}} = {2^{{2n} + n^{2}}{\prod\limits_{j = 1}^{n}\;\left( {2^{j} - 1} \right)}}$

and the number of circuit elements in the full Clifford circuit iscomputed using the following equation:

${{\mathcal{C}(n)}} = {2^{{2n} + n^{2}}{\prod\limits_{j = 1}^{n}\;\left( {4^{j} - 1} \right)}}$

where n is a number of qubits, j is an independent variable and j is anincrement from 1 to n number of qubits.

Another aspect of the present invention is to provide a classicalcomputer configured to execute a non-transitory computer-executablecode, the code when executed by the classical computer causes a quantumcomputer to: (a) provide randomly a plurality of Hadamard gates; (b)apply the randomly generated Hadamard gates to a plurality of qubits;(c) generate randomly a plurality of Hadamard-free Clifford circuits,wherein each of the plurality of Hadamard-free Clifford circuits isgenerated by at least randomly generating a uniformly distributed phase(P) gate, and randomly generating a uniformly distributed linear Booleaninvertible matrix of conditional NOT (CNOT) gate, and combining the Pand CNOT gates to form each of the plurality of Hadamard-free Cliffordcircuits; (d) combine each of the generated plurality of Hadamard-freeClifford circuits with corresponding each of the randomly generatedplurality of Hadamard gates to form a sequence of alternatingHadamard-free Clifford-Hadamard pairs circuit to form the randomizedbenchmarking protocol; and (e) measure noise in a quantum mechanicalprocessor using the randomized benchmarking protocol.

BRIEF DESCRIPTION OF THE DRAWINGS

The present disclosure, as well as the methods of operation andfunctions of the related elements of structure and the combination ofparts and economies of manufacture, will become more apparent uponconsideration of the following description and the appended claims withreference to the accompanying drawings, all of which form a part of thisspecification, wherein like reference numerals designate correspondingparts in the various figures. It is to be expressly understood, however,that the drawings are for the purpose of illustration and descriptiononly and are not intended as a definition of the limits of theinvention.

FIG. 1 is a flow chart of a method of generating a randomizedbenchmarking protocol, according to an embodiment of the presentinvention; and

FIG. 2 is a plot showing a comparison between a runtime of the presentmethod of generating a randomized benchmarking protocol and a runtime ofa conventional method of generating a randomized benchmarking protocolas a function of a number of qubits, according to an embodiment of thepresent invention.

DETAILED DESCRIPTION

In the following paragraphs, the term “randomized benchmarking” or“randomized benchmarking protocol” (RB) is used broadly to refer to anymethod for assessing the capabilities of quantum computing hardwareplatforms including quantum processors through estimating the averageerror rates that are measured under the implementation of long sequencesof random quantum gate operations. Randomized benchmarking protocols areused for verifying and validating quantum operations and can be used forthe optimization of quantum control procedures. Randomized benchmarkingis used to test the validity of quantum operations, which in turn isused to improve the functionality of the quantum computer hardware.

Other methods can be used for determining error rate in quantumcomputing hardware platforms. For example, another method fordetermining the error behavior of a gate implementation is to performprocess tomography. However, standard process tomography is limited byerrors in state preparation, measurement and one-qubit gates. Tomographysuffers from inefficient scaling with number of qubits and does notdetect adverse error-compounding when gates are composed in longsequences. An additional problem is that desirable error probabilitiesfor scalable quantum computing are of the order of 0.0001 or lower.Experimentally proving such low errors is challenging. Therefore,Randomized benchmarking is often preferred to the tomography methods. Arandomized benchmarking method can provide estimates of thecomputationally relevant errors or noise without relying on accuratestate preparation and measurement. Randomized benchmarking involves longsequences of randomly chosen gates. As a result, randomized benchmarkingcan also be used to verify that error behavior or noise is stable whenused in long computations.

FIG. 1 is a flow chart of a method of generating a randomizedbenchmarking protocol, according to an embodiment of the presentinvention. The method includes providing a randomly generated pluralityof Hadamard (H) gates, at 100, and applying the randomly generatedHadamard gates to a plurality of qubits, at 102. The method alsoincludes generating randomly a plurality of Hadamard-free Cliffordcircuits, at 104. Each of the plurality of Hadamard-free Cliffordcircuits is generated by at least randomly generating a uniformlydistributed phase (P) gate, and randomly generating a uniformlydistributed linear Boolean invertible matrix of conditional NOT (CNOT)gate, and combining the P and CNOT gates to form each of the pluralityof Hadamard-free Clifford circuits. The method further includescombining each of the generated plurality of Hadamard-free Cliffordcircuits with corresponding each of the randomly generated plurality ofHadamard gates to form a sequence of alternating Hadamard-freeClifford-Hadamard pairs circuit to form the randomized benchmarkingprotocol, at 106. The method also includes measuring noise in a quantummechanical processor using the randomized benchmarking protocol, at 108.

In an embodiment, randomly generating the uniformly distributed linearBoolean invertible matrix of conditional NOT (CNOT) gate includesrandomly generating a uniformly distributed linear Boolean invertiblematrix (a circuit with the CNOT gates).

In an embodiment, combining each of the generated plurality ofHadamard-free Clifford circuits with corresponding each of the randomlygenerated plurality of Hadamard gates to form a sequence of alternatingHadamard-free Clifford-Hadamard pairs circuit to form the randomizedbenchmarking protocol includes alternating stages of Hadamard layers andHadamard-free Clifford layers to form the modified randomizedbenchmarking protocol.

In an embodiment, generating randomly the plurality of Hadamard gatesincludes generating the plurality of Hadamard gates H_(1 . . . k). Wherek is an integer greater than 2 and corresponds to a number of randomqubits sampled from n-qubits, n being greater than or equal to 2.

In an embodiment, generating randomly the plurality of Hadamard-freeClifford circuits (layers) includes generating the plurality ofHadamard-free Clifford circuits F_(1 . . . k). Where k is an integergreater than 2 and corresponds to a number of random qubits sampled fromn-qubits, n being greater than or equal to 2.

In an embodiment, combining each of the generated plurality ofHadamard-free Clifford circuits with corresponding each of the randomlygenerated plurality of Hadamard gates to form the sequence ofalternating Hadamard-free Clifford-Hadamard pairs circuit includesforming the sequence F₁H₁ . . . F_(k)H_(k)F_(k+1)H_(k+1), where k is aninteger.

In an embodiment, the plurality of Hadamard-free Clifford circuits aredifferent from each other and are chosen at random according to aspecified distribution.

In an embodiment, generating each of the plurality of Hadamard-freeClifford circuits includes randomly generating the uniformly distributedphase (P) gate, randomly generating the uniformly distributed linearBoolean invertible matrix of conditional NOT (CNOT) gate, and randomlygenerating a uniformly distributed conditional rotation around axis-Z(CZ) gate, and combining the P, CZ, and CNOT gates to form each of theplurality of Hadamard-free Clifford circuits.

In an embodiment, combining each of the generated plurality ofHadamard-free Clifford circuits with corresponding each of the randomlygenerated plurality of Hadamard gates to form a sequence of alternatingHadamard-free Clifford-Hadamard pairs circuit includes forming thesequence F₁H₁ . . . F_(k)H_(k)F_(k+1)H_(k+1), where k is an integer. TheHadamard-free Clifford circuits F₁, F₂, . . . , F_(k) are equal to aproduct P.CZ.CNOT, where P is a uniformly distributed phase (P) gate, CZis a uniformly distributed conditional rotation around axis-Z (CZ) gate,and CNOT is a uniformly distributed linear Boolean conditional NOT(CNOT) circuit. The Hadamard-free Clifford circuit F_(k+1) is reduced tocontain no more than nk−k(k+1)/2 two-qubit CNOT and CZ gates. Where k isthe number of Hadamard gates in Bruhat expansion of F₁H₁ . . .F_(k)H_(k), where nk−k(k+1)/2 is obtained by reducing a left-handHadamard-free element by commuting and merging gates in it to the right.

In an embodiment, a computational complexity of generating therandomized benchmarking protocol grows quadratically with a number ofqubits.

In an embodiment, measuring the noise in the quantum mechanical circuitincludes combining an additional Hadamard-free Clifford-Hadamard pair tothe formed sequence of Hadamard-free Clifford-Hadamard pairs circuit toperform an inversion of a unitary transformation obtained by the formedsequence of Hadamard-free Clifford-Hadamard pairs circuit.

In an embodiment, the method also includes optimizing a length of therandomized benchmarking protocol by using fewer gates in the pluralityof Hadamard-free Clifford circuits than a number of circuit elements ina full Clifford circuit.

In an embodiment, elements of the plurality of Hadamard-free Cliffordcircuits are computed using the following equation:

${{\mathcal{F}(n)}} = {2^{{2n} + n^{2}}{\prod\limits_{j = 1}^{n}\;\left( {2^{j} - 1} \right)}}$and the number of circuit elements in the full Clifford circuit iscomputed using the following equation:

${{\mathcal{C}(n)}} = {2^{{2n} + n^{2}}{\prod\limits_{j = 1}^{n}\;\left( {4^{j} - 1} \right)}}$where n is a number of qubits, j is an independent variable and j is anincrement from 1 to n number of qubits.

In an embodiment, generating the uniformly distributed linear Booleaninvertible conditional NOT (CNOT) gate includes generating a lowertriangular random matrix (A) having random qubits 0 and/or 1; generatingan upper triangular random matrix (T) having random qubits 0 and/or 1;and multiplying the lower triangular random matrix (A) and the uppertriangular random matrix (T) to generate the uniformly distributedlinear Boolean invertible (CNOT) gate (CNOT=A·T).

In an embodiment, the uniformly distributed linear Boolean invertible(CNOT) gate can be generated by implementing the following Python code,for example.

import numpy as np n = 3 # matrix size A = np.random.randint(2, size=(n,n)) T = np.zeros((n,n),dtype=int) rows = list(range(n)) print (‘rows =’,rows) print (“”) for j in range(n−1):  v =np.random.randint(2,size=(1,n−j)) # pick random uniform nonzero n-bitstring  while np.count_nonzero(v)==0: v =np.random.randint(2,size=(1,n−j))  r = np.nonzero(v)[1][0] # let r bethe first nonzero element of v  A[j,rows] = 0; A[j,rows[r]] = 1 # assignvalues to A  for i in range(n−j): # assign values to TT[rows[r],rows[i]] = v[0,i]  del rows[r] A[n−1,rows[0]] = 1T[rows[0],rows[0]] = 1 U = np.matmul(A,T) % 2 # compute the randomuniform invertible matrix print(U)

Another aspect of the present invention is to provide a computerreadable medium on which is stored non-transitory computer-executablecode, which when executed by a classical computer causes a quantumcomputer to:

-   -   1) provide randomly a plurality of Hadamard gates;    -   2) apply the randomly generated Hadamard gates to a plurality of        qubits;    -   3) generate randomly a plurality of Hadamard-free Clifford        circuits, wherein each of the plurality of Hadamard-free        Clifford circuits is generated by at least randomly generating a        uniformly distributed phase (P) gate, and randomly generating a        uniformly distributed linear Boolean invertible matrix of        conditional NOT (CNOT) gate, and combining the P and CNOT gates        to form each of the plurality of Hadamard-free Clifford        circuits;    -   4) combine each of the generated plurality of Hadamard-free        Clifford circuits with corresponding each of the randomly        generated plurality of Hadamard gates to form a sequence of        alternating Hadamard-free Clifford-Hadamard pairs circuit to        form the randomized benchmarking protocol; and    -   5) measure noise in a quantum mechanical processor using the        randomized benchmarking protocol.

In an embodiment, the non-transitory computer-executable code whenexecuted by the classical computer causes the quantum computer togenerate the plurality of Hadamard gates H_(1 . . . k), where k is aninteger greater than 2 and corresponds to a number of random qubitssampled from n-qubits, n being greater than or equal to 2.

In an embodiment, the non-transitory computer-executable code whenexecuted by the classical computer causes the quantum computer togenerate the plurality of Hadamard-free Clifford circuits F_(1 . . . k),where k is an integer greater than 2 and corresponds to a number ofrandom qubits sampled from n-qubits, n being greater than or equal to 2.

In an embodiment, the non-transitory computer-executable code whenexecuted by the classical computer causes the quantum computer tocombine each of the generated plurality of Hadamard-free Cliffordcircuits with corresponding each of the randomly generated plurality ofHadamard gates to form the sequence F₁H₁ . . . F_(k)H_(k)F_(k+1)H_(k+1),where k is an integer.

In an embodiment, the plurality of Hadamard-free Clifford circuits aredifferent from each other and are chosen at random according to aspecified distribution.

In an embodiment, the non-transitory computer-executable code whenexecuted by the classical computer causes the quantum computer to applythe uniformly distributed phase (P) gate, apply the uniformlydistributed linear Boolean invertible matrix of conditional NOT (CNOT)gate, and apply a uniformly distributed conditional rotation aroundaxis-Z (CZ) gate, and combine the P, CZ, and CNOT gates to form each ofthe plurality of Hadamard-free Clifford circuits.

In an embodiment, the non-transitory computer-executable code whenexecuted by the classical computer causes the quantum computer tooptimize a length of the randomized benchmarking protocol by using fewergates in the plurality of Hadamard-free Clifford circuits than a numberof circuit elements in a full Clifford circuit.

In an embodiment, elements of the plurality of Hadamard-free Cliffordcircuits are computed using the following equation:

${{\mathcal{F}(n)}} = {2^{{2n} + n^{2}}{\prod\limits_{j = 1}^{n}\;\left( {2^{j} - 1} \right)}}$and the number of circuit elements in the full Clifford circuit iscomputed using the following equation:

${{\mathcal{C}(n)}} = {2^{{2n} + n^{2}}{\prod\limits_{j = 1}^{n}\;\left( {4^{j} - 1} \right)}}$where n is a number of qubits, j is an independent variable and j is anincrement from 1 to n number of qubits.

In an embodiment, to synthesize a random element of group F, take randomBoolean 2n-tuple to set Phase (P) gates, n(n−1)/2 random Boolean valuesto set CZ gates, and use Randall's algorithm to synthesize randominvertible Boolean matrix using n²+O(1) random bits and computationalcomplexity O(n^(2.373)). When a circuit representation suffices, thecomplexity decreases to O(n²). The random Clifford unitary generationuses 2n²+O(n) random bits, whereas the present method uses 1.5n²+O(n)random bits. In an embodiment, a computational complexity of the presentmethod is O(n^(2.373)) or O(n²), depending on the desired datastructure, whereas a computational complexity of a conventional methodis O(n³).

In an embodiment, the present method returns data structures (phasepolynomial, or circuit) that can be more useful in the context of RBthan the symplectic matrices used in conventional methods. Furthermore,in an embodiment, the present method uses fewer random bits (by a factor¾), which is calculated by dividing 1.5n²+O(n) random bits by 2n²+O(n)random bits. In addition, the computational complexity of the presentmethod is significantly lower.

FIG. 2 is a plot showing a comparison between a runtime of the presentmethod of generating a randomized benchmarking protocol and a runtime ofa conventional method of generating a randomized benchmarking protocolas a function of a number of qubits, according to an embodiment of thepresent invention. The horizontal axis corresponds to the number ofqubits and the vertical axis corresponds to the runtime. Dots 202 in theplot correspond to the runtime of a first method requiring that theClifford unitary be generated in the form P.CZ.CNOT. It is noted thatthe complexity of the implementation scales cubically in n (this said,it can be improved to O(n^(2.3729)) with the use of advanced matrixmultiplication algorithms), but is still a lot more efficient than dots206, showing the state of the art, per Koenig and Smolin. Dots 204correspond to the runtime in the scenario when the answer can be givenin a circuit or phase polynomial form using a second method according toan embodiment of the present invention. The complexity of generatingdots 202 is quadratic. As such, for larger n, (not shown) the gap inefficiency of 202 (the first method) compared to 204 (the second method)will continue widening, with 202 (the first method) being moreefficient. Dots 206 in the plot correspond to the runtime of aconventional method (Koenig-Smolin method) of generating the Cliffordunitary circuit. As it is clearly shown in this plot, the runtime of thefirst and second methods (dots 202 and dots 204, respectively) ofgenerating the random Clifford unitary circuit according to embodimentsof the present invention is about 1000-fold lower that the runtime ofthe conventional method of generating the random Clifford unitarycircuit. Therefore, the method(s) of a method of generating a randomizedbenchmarking protocol according to embodiments of the present inventionis generally faster than conventional methods of generating a randomizedbenchmarking protocol.

As it can be appreciated from the above paragraphs, another aspect ofthe present invention is to provide a classical computer configured toexecute a non-transitory computer-executable code, the code whenexecuted by the classical computer causes a quantum computer to:

-   -   1) provide a randomly generated plurality of Hadamard gates;    -   2) apply the randomly generated Hadamard gates to a plurality of        qubits;    -   3) generate randomly a plurality of Hadamard-free Clifford        circuits, wherein each of the plurality of Hadamard-free        Clifford circuits is generated by at least randomly generate a        uniformly distributed phase (P) gate, and randomly generating a        uniformly distributed linear Boolean invertible matrix of        conditional NOT (CNOT) gate, and combining the P and CNOT gates        to form each of the plurality of Hadamard-free Clifford        circuits;    -   4) combine each of the generated plurality of Hadamard-free        Clifford circuits with corresponding each of the randomly        generated plurality of Hadamard gates to form a sequence of        alternating Hadamard-free Clifford-Hadamard pairs circuit to        form the randomized benchmarking protocol; and    -   5) measure noise in a quantum mechanical processor using the        randomized benchmarking protocol.

In an embodiment, the code may be stored in a computer program productwhich include a computer readable medium or storage medium or media.Examples of suitable storage medium or media include any type of diskincluding floppy disks, optical disks, DVDs, CD ROMs, magnetic opticaldisks, RAMs, EPROMs, EEPROMs, magnetic or optical cards, hard disk,flash card (e.g., a USB flash card), PCMCIA memory card, smart card, orother media. In another embodiment, the code can be downloaded from aremote conventional or classical computer or server via a network suchas the internet, an ATM network, a wide area network (WAN) or a localarea network. In yet another embodiment, the code can reside in the“cloud” on a server platform, for example. In some embodiments, the codecan be embodied as program products in the conventional or classicalcomputer such as a personal computer or server or in a distributedcomputing environment including a plurality of computers that interactswith the quantum computer by sending instructions to and receiving datafrom the quantum computer.

Generally, the classical or conventional computer provides inputs andreceives outputs from the quantum computer. The inputs may includeinstructions included as part of the code. The outputs may includequantum data results of a computation of the code on the quantumcomputer.

The classical computer interfaces with the quantum computer via aquantum computer input interface and a quantum computer outputinterface. The classical computer sends commands or instructionsincluded within the code to the quantum computer system via the inputand the quantum computer returns outputs of the quantum computation ofthe code to the classical computer via the output. The classicalcomputer can communicate with the quantum computer wirelessly or via theinternet. In an embodiment, the quantum computer can be a quantumcomputer simulator simulated on a classical computer. For example, thequantum computer simulating the quantum computing simulator can be oneand the same as the classical computer. In another embodiment, thequantum computer is a superconducting quantum computer. In anembodiment, the superconducting quantum computer includes one or morequantum circuits (Q chips), each quantum circuit comprises a pluralityof qubits, one or more quantum gates, measurement devices, etc.

The descriptions of the various embodiments of the present inventionhave been presented for purposes of illustration, but are not intendedto be exhaustive or limited to the embodiments disclosed. Manymodifications and variations will be apparent to those of ordinary skillin the art without departing from the scope and spirit of the describedembodiments. The terminology used herein was chosen to best explain theprinciples of the embodiments, the practical application or technicalimprovement over technologies found in the marketplace, or to enableothers of ordinary skill in the art to understand the embodimentsdisclosed herein.

We claim:
 1. A method of generating a randomized benchmarking protocolcomprising: providing a randomly generated plurality of Hadamard gates;applying the randomly generated Hadamard gates to a plurality of qubits;generating randomly a plurality of Hadamard-free Clifford circuits,wherein each of the plurality of Hadamard-free Clifford circuits isgenerated by at least randomly generating a uniformly distributed phase(P) gate, and randomly generating a uniformly distributed linear Booleaninvertible matrix of conditional NOT (CNOT) gate, and combining the Pand CNOT gates to form each of the plurality of Hadamard-free Cliffordcircuits; combining each of the generated plurality of Hadamard-freeClifford circuits with corresponding each of the randomly generatedplurality of Hadamard gates to form a sequence of alternatingHadamard-free Clifford-Hadamard pairs circuit to form the randomizedbenchmarking protocol; and measuring noise in a quantum mechanicalprocessor using the randomized benchmarking protocol.
 2. The methodaccording to claim 1, wherein generating randomly the plurality ofHadamard gates comprises generating the plurality of Hadamard gatesH_(1 . . . k), where k is an integer greater than 2 and corresponds to anumber of random qubits sampled from n-qubits, n being greater than orequal to
 2. 3. The method according to claim 1, wherein generatingrandomly the plurality of Hadamard-free Clifford circuits comprisesgenerating the plurality of Hadamard-free Clifford circuitsF_(1 . . . k), where k is an integer greater than 2 and corresponds to anumber of random qubits sampled from n-qubits, n being greater than orequal to
 2. 4. The method according to claim 1, wherein combining eachof the generated plurality of Hadamard-free Clifford circuits withcorresponding each of the randomly generated plurality of Hadamard gatesto form the sequence of alternating Hadamard-free Clifford-Hadamardpairs circuit comprises forming the sequence F₁H₁ . . .F_(k)H_(k)F_(k+1)H_(k+1), where k is an integer.
 5. The method accordingto claim 1, wherein the plurality of Hadamard-free Clifford circuits aredifferent from each other and are chosen at random according to aspecified distribution.
 6. The method according to claim 1, whereingenerating each of the plurality of Hadamard-free Clifford circuitscomprises randomly generating the uniformly distributed phase (P) gate,randomly generating the uniformly distributed linear Boolean invertiblematrix of conditional NOT (CNOT) gate, and randomly generating auniformly distributed conditional rotation around axis-Z (CZ) gate, andcombining the P, CZ, and CNOT gates to form each of the plurality ofHadamard-free Clifford circuits.
 7. The method according to claim 1,wherein combining each of the generated plurality of Hadamard-freeClifford circuits with corresponding each of the randomly generatedplurality of Hadamard gates to form a sequence of Hadamard-freeClifford-Hadamard pairs circuit comprises forming the sequence F₁H₁ . .. F_(k)H_(k)F_(k+1)H_(k+1), where k is an integer, wherein: theHadamard-free Clifford circuits F₁, F₂, . . . , F_(k) are equal to aproduct P.CZ.CNOT, where P is a uniformly distributed phase (P) gate, CZis a uniformly distributed conditional rotation around axis-Z (CZ) gate,and CNOT is a uniformly distributed linear Boolean conditional NOT(CNOT) circuit, and the Hadamard-free Clifford circuit F_(k+1) isreduced to contain no more than nk−k(k+1)/2 two-qubit CNOT and CZ gates,where k is the number of Hadamard gates in Bruhat expansion of F₁H₁ . .. F_(k)H_(k), where nk−k(k+1)/2 is obtained by reducing a left-handHadamard-free element by commuting and merging gates in it to the right.8. The method according to claim 1, wherein a computational complexityof generating the randomized benchmarking protocol grows quadraticallywith a number of qubits.
 9. The method according to claim 1, whereinmeasuring the noise in the quantum mechanical circuit comprisescombining an additional Hadamard-free Clifford-Hadamard pair to theformed sequence of Hadamard-free Clifford-Hadamard pairs circuit toperform an inversion of a unitary transformation obtained by the formedsequence of Hadamard-free Clifford-Hadamard pairs circuit.
 10. Themethod according to claim 1, further comprising optimizing a length ofthe randomized benchmarking protocol by using fewer gates in theplurality of Hadamard-free Clifford circuits than a number of circuitelements in a full Clifford circuit.
 11. The method according to claim10, wherein a size of the plurality of Hadamard-free Clifford circuitsare computed using the following equation:${{\mathcal{F}(n)}} = {2^{{2n} + n^{2}}{\prod\limits_{j = 1}^{n}\;\left( {2^{j} - 1} \right)}}$and the number of circuit elements in the full Clifford circuit iscomputed using the following equation:${{\mathcal{C}(n)}} = {2^{{2n} + n^{2}}{\prod\limits_{j = 1}^{n}\;\left( {4^{j} - 1} \right)}}$where n is a number of qubits, j is an independent variable and j is anincrement from 1 to n number of qubits.
 12. A computer readable mediumon which is stored non-transitory computer-executable code, which whenexecuted by a classical computer causes a quantum computer to: provide arandomly generated plurality of Hadamard gates; apply the randomlygenerated Hadamard gates to a plurality of qubits; generate randomly aplurality of Hadamard-free Clifford circuits, wherein each of theplurality of Hadamard-free Clifford circuits is generated by at leastrandomly generating a uniformly distributed phase (P) gates, andrandomly generating a uniformly distributed linear Boolean invertiblematrix of conditional NOT (CNOT) gates, and combining the P and CNOTgates to form each of the plurality of Hadamard-free Clifford circuits;combine each of the generated plurality of Hadamard-free Cliffordcircuits with corresponding each of the randomly generated plurality ofHadamard gates to form a sequence of alternating Hadamard-freeClifford-Hadamard pairs circuit to form the randomized benchmarkingprotocol; and measure noise in a quantum mechanical processor using therandomized benchmarking protocol.
 13. The computer readable mediumaccording to claim 12, wherein the non-transitory computer-executablecode when executed by the classical computer causes the quantum computerto generate the plurality of Hadamard gates H_(1 . . . k), where k is aninteger greater than 2 and corresponds to a number of random qubitssampled from n-qubits, n being greater than or equal to
 2. 14. Thecomputer readable medium according to claim 12, wherein thenon-transitory computer-executable code when executed by the classicalcomputer causes the quantum computer to generate the plurality ofHadamard-free Clifford circuits F_(1 . . . k), where k is an integergreater than 2 and corresponds to a number of random qubits sampled fromn-qubits, n being greater than or equal to
 2. 15. The computer readablemedium according to claim 12, wherein the non-transitorycomputer-executable code when executed by the classical computer causesthe quantum computer to combine each of the generated plurality ofHadamard-free Clifford circuits with corresponding each of the randomlygenerated plurality of Hadamard gates to form the sequence F₁H₁ . . .F_(k)H_(k)F_(k+1)H_(k+1), where k is an integer.
 16. The methodaccording to claim 12, wherein the plurality of Hadamard-free Cliffordcircuits are different from each other and are chosen at randomaccording to a specified distribution.
 17. The computer readable mediumaccording to claim 12, wherein the non-transitory computer-executablecode when executed by the classical computer causes the quantum computerto apply the uniformly distributed phase (P) gates, apply the uniformlydistributed random linear Boolean invertible matrix computed byconditional NOT (CNOT) gates, and apply a random uniformly distributedconditional rotation around axis-Z (CZ) gate, and combine the P, CZ, andCNOT gates to form each of the plurality of Hadamard-free Cliffordcircuits.
 18. The computer readable medium according to claim 12,wherein the non-transitory computer-executable code when executed by theclassical computer causes the quantum computer to optimize a length ofthe randomized benchmarking protocol by using fewer gates in theplurality of Hadamard-free Clifford circuits than a number of circuitelements in a full Clifford circuit.
 19. The computer readable mediumaccording to claim 18, wherein the number of elements of the pluralityof Hadamard-free Clifford circuits are computed using the followingequation:${{\mathcal{F}(n)}} = {2^{{2n} + n^{2}}{\prod\limits_{j = 1}^{n}\;\left( {2^{j} - 1} \right)}}$and the number of circuit elements in the full Clifford circuit iscomputed using the following equation:${{\mathcal{C}(n)}} = {2^{{2n} + n^{2}}{\prod\limits_{j = 1}^{n}\;\left( {4^{j} - 1} \right)}}$where n is a number of qubits, j is an independent variable and j is anincrement from 1 to n number of qubits.
 20. A classical computerconfigured to execute a non-transitory computer-executable code, thecode when executed by the classical computer causes a quantum computerto: provide randomly a plurality of Hadamard gates; apply the randomlygenerated Hadamard gates to a plurality of qubits; generate randomly aplurality of Hadamard-free Clifford circuits, wherein each of theplurality of Hadamard-free Clifford circuits is generated by at leastrandomly generating a uniformly distributed phase (P) gate, and randomlygenerating a uniformly distributed linear Boolean invertible matrix ofconditional NOT (CNOT) gate, and combining the P and CNOT gates to formeach of the plurality of Hadamard-free Clifford circuits; combine eachof the generated plurality of Hadamard-free Clifford circuits withcorresponding each of the randomly generated plurality of Hadamard gatesto form a sequence of alternating Hadamard-free Clifford-Hadamard pairscircuit to form the randomized benchmarking protocol; and measure noisein a quantum mechanical processor using the randomized benchmarkingprotocol.